Asymptotics of particle trajectories in infinite one-dimensional systems with collisions

We investigate the asymptotic behavior of the trajectories of a tagged particle (tp) in an infinite one-dimensional system of point particles. The particles move independently when not in contact: the only interactions are Harris type generalized elastic collisions which prevent crossings. This is achieved by relabeling the independent trajectories when they cross. When these trajectories are differentiable, as in particles with velocities undergoing Ornstein-Uhlenbeck processes, collisions correspond to exchange of velocities. We prove very generally that the suitably scaled tp trajectory converges (weakly) to a simple Gaussian process. This extends the results of Spitzer for New tonian particles to very general non-crossing processes. The proof is based on the consideration of the simpler process which counts the crossings of the origin by the independent trajectories.     

A particle method to solve the Navier-Stokes system

Summary We extend to the case of the two-dimensional Navier-Stokes equations, a particle method introduced in a previous paper to solve linear convection-diffusion equations. The method is based on a viscous splitting of the operator. The particles move under the effect of the velocity field but are not affected by the diffusion which is taken into account by the weights. We prove the stability and the convergence of the method.     

Continuity for the asymptotic shape in the frog model with random initial configurations

We consider the so-called frog model with random initial configurations, which is described by the following evolution mechanism of simple random walks on the multidimensional cubic lattice: Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and they independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start doing independent simple random walks. An interest of this model is how initial configurations affect the asymptotic shape of the set of all sites visited by active particles up to a certain time. Thus, in this paper, we prove continuity for the asymptotic shape in the law of the initial configuration.     

Nonequilibrium Density Profiles in Lorentz Tubes with Thermostated Boundaries

We consider a long Lorentz tube with absorbing boundaries. Particles are injected into the tube from the left end. We compute the equilibrium density profiles in two cases: the semi‐infinite tube (in which case the density is constant) and a long finite tube (in which case the density is linear). In the latter case, we also show that convergence to equilibrium is well described by the heat equation. In order to prove these results, we obtain new results for the Lorentz particle that are of independent interest. First, we show that a particle conditioned not to hit the boundary for a long time converges to the Brownian meander. Second, we prove several local limit theorems for particles having a prescribed behavior in the past. © 2016 Wiley Periodicals, Inc.     

A multidimensional nonlinear sixth-order quantum diffusion equation

This paper is concerned with the analysis of a sixth-order nonlinear parabolic equation whose solutions describe the evolution of the particle density in a quantum fluid. We prove the global-in-time existence of weak nonnegative solutions in two and three space dimensions under periodic boundary conditions. Moreover, we show that these solutions are smooth and classical whenever the particle density is strictly positive, and we prove the long-time convergence to the spatial homogeneous equilibrium at a universal exponential rate. Our analysis strongly uses the Lyapunov property of the entropy functional.     

A particle transport problem with non-homogeneous reflection boundary conditions

We consider particle transport in a three-dimensional convex region V, bounded by the regular surface ∂V. We assume that particles are specularly reflected by ∂V and that a source q is assigned on ∂V; more general non-homogeneous boundary conditions are also discussed. The problem is non-linear because the boundary condition is not homogeneous. We prove existence of a unique strict solution and by using the theory of semigroups we derive the explicit expression of such a solution in terms of the boundary source q. In the appendix, we indicate how some properties of affine operators can be used to derive the solution. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Time required to unify all particles in the scheme of equiprobable allocation into a sequence of cell layers

We study the scheme of equiprobable allocations of particles into a sequence of cell layers, where the particles put into the same cell are considered as a single particle. We present conditions under which there exist, with positive probability, nonunified particles at each of the layers. For the case in which the number of cells at each of the layers is equal to the number of original particles, we prove the limit theorem for the time instant at which all the particles are unified into a single particle.     

Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction

Summary For Gibbsian systems of particles inR ^d , we investigate large deviations of the translation invariant empirical fields in increasing boxes. The particle interaction is given by a superstable, regular pair potential. The large deviation principle is established for systems with free or periodic boundary conditions and, under a stronger stability hypothesis on the potential, for systems with tempered boundary conditions, and for tempered (infinite-volume) Gibbs measures. As a by-product we obtain the Gibbs variational formula for the pressure. We also prove the asymptotic equivalence of microcanonical and grand canonical Gibbs distributions and establish a variational expression for the thermodynamic entropy density.     

A generalization of Rado's Theorem for almost graphical boundaries

In this paper, we prove a generalization of Rado's Theorem, a fundamental result of minimal surface theory, which says that minimal surfaces over a convex domain with graphical boundaries must be disks which are themselves graphical. We will show that, for a minimal surface of any genus, whose boundary is ``almost graphical' in some sense, that the surface must be graphical once we move sufficiently far from the boundary.     

Two stages of motion of anharmonic oscillators modeling fast particles in crystals

We consider the evolution of the spatial distribution of fast atomic particles as they pass through a crystal. At small penetration depths where the particles move without significant loss of coherence, the joint probability density function becomes smoother at the expense of the gradient of the anharmonic potential of the planar channel. We show that as a result of this smoothing, there arises a quasiequilibrium (quasistationary) state of the subsystem of fast particles. The further evolution of the particle distribution is caused by nonelastic scattering and is described by the kinetic equation. At this stage, the anharmonic character of particle oscillations between the channel walls results in a renormalization of the total phonon scattering cross section of particles, and the renormalization is determined by the fourth-order anharmonic interaction.     

A Lagrangian simulation of particle deposition in a turbulent boundary layer in the presence of thermophoresis

Knowledge of particle deposition in turbulent flows is often required in engineering situations. Examples include fouling of turbine blades, plate-out in nuclear reactors and soot deposition. Thus it is important for numerical simulations to be able to predict particle deposition. Particle deposition is often principally determined by the forces acting on the particles in the boundary layer. The particle tracking facility in the CFD code uses the eddy lifetime model to simulate turbulent particle dispersion, no specific boundary layer being modelled. The particle tracking code has been modified to include a boundary layer. The non-dimensional yplus, y⁺, distance of the particle from the wall is determined and then values for the fluid velocity, fluctuating fluid velocity and eddy lifetime appropriate for a turbulent boundary layer used. Predictions including the boundary layer have been compared against experimental data for particle deposition in turbulent pipe flow. The results giving much better agreement. Many engineering problems also involve heat transfer and hence temperature gradients. Thermophoresis is a phenomena by which small particles experience a force in the opposite direction to the temperature gradient. Thus particles will tend to deposit on cold walls and be repulsed by hot walls. The effect of thermophoresis on the deposition of particles can be significant. The modifications of the particle tracking facility have been extended to include the effect of thermophoresis. A preliminary test case involving the deposition of particles in a heated pipe has been simulated. Comparison with experimental data from an extensive experimental programme undertaken at ISPRA, known as STORM (Simplified Tests on Resuspension Mechanisms), has been made.     

Spectral Problems for the Dirac System with Spectral Parameter in Local Boundary Conditions

We consider a spectral boundary value problem in a 3-dimensional bounded domain for the Dirac system that describes the behavior of a relativistic particle in an electromagnetic field. The spectral parameter is contained in a local boundary condition. We prove that the eigenvalues of the problem have finite multiplicities and two points of accumulation, zero and infinity and indicate the asymptotic behavior of the corresponding series of eigenvalues. We also show the existence of an orthonormal basis on the boundary consisting of two-dimensional parts of the four-dimensional eigenfunctions.     

Existence of an infinite particle limit of stochastic ranking process

We study a stochastic particle system which models the time evolution of the ranking of books by online bookstores (e.g., Amazon.co.jp). In this system, particles are lined in a queue. Each particle jumps at random jump times to the top of the queue, and otherwise stays in the queue, being pushed toward the tail every time another particle jumps to the top. In an infinite particle limit, the random motion of each particle between its jumps converges to a deterministic trajectory. (This trajectory is actually observed in the ranking data on web sites.) We prove that the (random) empirical distribution of this particle system converges to a deterministic space–time-dependent distribution. A core of the proof is the law of large numbers for dependent random variables.     

Ideal gas approximation for a two-dimensional rarefied gas under Kawasaki dynamics

In this paper we consider a two-dimensional lattice gas under Kawasaki dynamics, i.e., particles hop around randomly subject to hard-core repulsion and nearest-neighbor attraction. We show that, at fixed temperature and in the limit as the particle density tends to zero, such a gas evolves in a way that is close to an ideal gas, where particles have no interaction. In particular, we prove three theorems showing that particle trajectories are non-superdiffusive and have a diffusive spread-out property. We also consider the situation where the temperature and the particle density tend to zero simultaneously and focus on three regimes corresponding to the stable, the metastable and the unstable gas, respectively.     

On the chaotic behavior of non-flat billiards

We study the problem of the motion of a particle on a non-flat billiard. The particle is subject to the gravity and to a small amplitude periodic (or almost periodic) forcing and is reflected with respect to the normal axis when it hits the boundary of the billiard. We prove that the unperturbed problem has an impact homoclinic orbit and give a Melnikov type condition so that the perturbed problem exhibit chaotic behavior in the sense of Smale’s horseshoe.     

Diffusion of color in the simple exclusion process

We prove a diffusion scaling limit for the macroscopic densities of colored particles performing the simply excluded random walk, and relate this to the limiting behavior of a test particle in equilibrium.     

A tagged particle process in the Boltzmann-Grad limit for the Broadwell modell

Summary We study a tagged particle process for a model dynamical system in which identical particles move deterministically with discrete velocities, initially starting from a random configuration. We pass to the Boltzmann-Grad limit so that the tagged particle process converges to a nontrivial process (for short times). We can show that recollisions are vanishing in this limit, and this fact may have one expect that the limiting process would be Markovian. Nevertheless it is not Markovian, for which claim we give intuitive reasoning as well as a mathematical proof.Supported in part by Grant-in-Aid for Scientific Research (No. 62302006), Ministry of Education, Science and Culture     

One-dimensional free boundary problem for actin-based propulsion of Listeria

Some bacteria move inside cells by recruiting the actin filaments of the host cells. The filaments are polymerized at the back surface of the bacteria, and they move away, forming a “comet” tail behind the bacterium, which consists of gel network. We develop a one-dimensional mathematical model of the gel based on partial differential equations which involve the number of filaments, the density and velocity of the gel, and the pressure. The two end-points of the gel form two free boundaries. The resulting free boundary problem is rather non-standard. We prove local existence and uniqueness.     

On the mathematics of emergence

We describe a setting where convergence to consensus in a population of autonomous agents can be formally addressed and prove some general results establishing conditions under which such convergence occurs. Both continuous and discrete time are considered and a number of particular examples, notably the way in which a population of animals move together, are considered as particular instances of our setting. This article is based on the 1st Takagi Lectures that the second author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006. Steve Smale Partially supported by an NSF grant.